#373871

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1248 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(2,3)(5,6,11,16,12,17,10,29)(7,18,8,22,21,14,20,13)(9,24,27,33,28,36,26,15)(19,34,23,31,35,30,25,32);(1,2,4,3)(5,7,10,20,12,21,11,8)(6,13,16,18,17,22,29,14)(9,19,26,25,28,35,27,23)(15,30,24,32,33,34,36,31);(5,8,11,21,12,20,10,7)(6,14,16,13,17,18,29,22)(9,25,27,19,28,23,26,35)(15,32,24,34,33,31,36,30);(1,3,4,2)(5,9,10,26,12,28,11,27)(6,15,16,24,17,33,29,36)(7,19,20,25,21,35,8,23)(13,30,18,32,22,34,14,31)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=h,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=di,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280138, #72280451, #72280560, …
maximal normal subgroup of	#72280138, #72280451, #72280560, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281067);